Parametric tilings and scattered data approximation Michael
Floater. International journal of Shape Modeling 4 (1998) pg. 165-187.
This paper extends the work of his '97 paper to polygons (as opposed
to triangles). It also relates this work to the Harmonic map of Eck, et al .
Proofs in paper
- If your mesh has > 3 sided polygons, the convex combination approach still works.
- The shape preserving weights, introduced in the 97 paper, will reproduce all tilings, i.e., given a mesh that's already in the plane, you'll get out the same mesh.
- Choosing weights by the least squares approach (putting weights
on the edges) will not always reproduce a tiling.
- The harmonic map introduced by Eck is, indeed, a harmonic map. As
long as the weights are well-defined then it will reproduce
tilings. Hence, it ends up being similar to the shape preserving
approach, in these cases. [Remember that the harmonic map does not
make any guarantees about folding.]